Translation plane

In mathematics, a translation plane is a projective plane which admits a certain group of symmetries (described below). Along with the Hughes planes and the Figueroa planes, translation planes are among the most well-studied of the known non-Desarguesian planes, and the vast majority of known non-Desarguesian planes are either translation planes, or can be obtained from a translation plane via successive iterations of dualization and/or derivation.[1]

In a projective plane, let P represent a point, and l represent a line. A central collineation with center P and axis l is a collineation fixing every point on l and every line through P. It is called an elation if P is on l, otherwise it is called a homology. The central collineations with centre P and axis l form a group.[2] A line l in a projective plane Π is a translation line if the group of all elations with axis l acts transitively on the points of the affine plane obtained by removing l from the plane Π, Π_l (the affine derivative of Π). A projective plane with a translation line is called a translation plane.

The affine plane obtained by removing the translation line is called an affine translation plane. While it is often easier to work with projective planes, in this context several authors use the term translation plane to mean affine translation plane.[3][4]

Algebraic construction with coordinates

Every projective plane can be coordinatized by at least one planar ternary ring.[5] For translation planes, it is always possible to coordinatize with a quasifield.[6] However, some quasifields satisfy additional algebraic properties, and the corresponding planar ternary rings coordinatize translation planes which admit additional symmetries. Some of these special classes are:

• Nearfield planes - coordinatized by nearfields.
• Semifield planes - coordinatized by semifields, semifield planes have the property that their dual is also a translation plane.
• Moufang planes - coordinatized by alternative division rings, Moufang planes are exactly those translation planes that have at least two translation lines. Every finite Moufang plane is Desarguesian and every Desarguesian plane is a Moufang plane, but there are infinite Moufang planes that are not Desarguesian (such as the Cayley plane).

Given a quasifield with operations + (addition) and ${\displaystyle \cdot }$ (multiplication), one can define a planar ternary ring to create coordinates for a translation plane. However, it is more typical to create an affine plane directly from the quasifield by defining the points as pairs ${\displaystyle (a,b)}$ where ${\displaystyle a}$ and ${\displaystyle b}$ are elements of the quasifield, and the lines are the sets of points ${\displaystyle (x,y)}$ satisfying an equation of the form ${\displaystyle y=m\cdot x+b}$ , as ${\displaystyle m}$ and ${\displaystyle b}$ vary over the elements of the quasifield, together with the sets of points ${\displaystyle (x,y)}$ satisfying an equation of the form ${\displaystyle x=a}$ , as ${\displaystyle a}$ varies over the elements of the quasifield.[7]

Geometric construction with spreads

Translation planes are related to spreads of odd-dimensional projective spaces by the André/Bruck-Bose construction.[8][9] A spread of PG(2n+1, K), where ${\displaystyle n\geq 1}$ is an integer and K a division ring, is a partition of the space into pairwise disjoint n-dimensional subspaces. In the finite case, a spread of PG(2n+1, q) is a set of qⁿ⁺¹ + 1 n-dimensional subspaces, with no two intersecting.

Given a spread S of PG(2n +1, K), the André/Bruck-Bose construction produces a translation plane as follows: Embed PG(2n+1, K) as a hyperplane ${\displaystyle \Sigma }$ of PG(2n+2, K). Define an incidence structure A(S) with "points," the points of PG(2n+2, K) not on ${\displaystyle \Sigma }$ and "lines" the (n+1)-dimensional subspaces of PG(2n+2, K) meeting ${\displaystyle \Sigma }$ in an element of S. Then A(S) is an affine translation plane. In the finite case, this procedure produces a translation plane of order qⁿ⁺¹.

The converse of this statement is almost always true.[10] Any translation plane which is coordinatized by a quasifield that is finite-dimensional over its kernel K (K is necessarily a division ring) can be generated from a spread of PG(2n+1, K) using the André/Bruck-Bose construction, where (n+1) is the dimension of the quasifield, considered as a module over its kernel. An instant corollary of this result is that every finite translation plane can be obtained from this construction.

Reguli and regular spreads

Let ${\displaystyle \Sigma }$ be the projective space PG(2n+1, K) for ${\displaystyle n\geq 1}$ an integer, and K a division ring. A regulus[11] R in ${\displaystyle \Sigma }$ is a collection of pairwise disjoint n-dimensional subspaces with the following properties:

1. R contains at least 3 elements
2. Every line meeting three elements of R, called a transversal, meets every element of R
3. Every point of a transversal to R lies on some element of R

Any three pairwise disjoint n-dimensional subspaces in ${\displaystyle \Sigma }$ lie in a unique regulus.[12] A spread S of ${\displaystyle \Sigma }$ is regular if for any three distinct n-dimensional subspaces of S, all the members of the unique regulus determined by them are contained in S. For any division ring K with more than 2 elements, if a spread S of PG(2n+1, K) is regular, then the translation plane created by that spread via the André/Bruck-Bose construction is a Moufang plane. A slightly weaker converse holds: if a translation plane is Pappian, then it can be generated via the André/Bruck-Bose construction from a regular spread.[13]

In the finite case, K must be a field of order ${\displaystyle q>2}$, and the classes of Moufang, Desarguesian and Pappian planes are all identical, so this theorem can be refined to state that a spread S of PG(2n+1, q) is regular if and only if the translation plane created by that spread via the André/Bruck-Bose construction is Desarguesian.

All spreads of PG(2n+1, 2) are trivially regular, since a regulus only contains three elements. While the only translation plane of order 8 is Desarguesian, there are known to be non-Desarguesian translation planes of order 2^e for every integer ${\displaystyle e\geq 4}$.[14]

Families of non-Desarguesian translation planes

• Hall planes
• André planes
• Subregular planes
• Nearfield planes
• Semifield planes

Finite translation planes of small order

It is well known that the only projective planes of order 8 or less are Desarguesian, and there are no known non-Desarguesian planes of prime order.[15] Finite translation planes must have prime power order. There are four projective planes of order 9, of which two are translation planes: the Desarguesian plane, and the Hall plane. The following table details the current state of knowledge:

Order	Number of Non-Desarguesian Translation Planes
9	1
16	7[16][17]
25	20[18][19][20]
27	6[21][22]
32	≥8[23]
49	1346[24][25]
64	≥2833[26]

Algebraic representation

An algebraic representation of (affine) translation planes can be obtained as follows: Let V be a 2n-dimensional vector space over a field F. A spread of V is a set S of n-dimensional subspaces of V that partition the non-zero vectors of V. The members of S are called the components of the spread and if V_i and V_j are distinct components then V_i ⊕ V_j = V. Let A be the incidence structure whose points are the vectors of V and whose lines are the cosets of components, that is, sets of the form v + U where v is a vector of V and U is a component of the spread S. Then:[27]

A is an affine plane and the group of translations x → x + w for w in V is an automorphism group acting regularly on the points of this plane.

Finite construction

Let F = GF(q) = F_q, the finite field of order q and V the 2n-dimensional vector space over F represented as:

${\displaystyle V=\{(x,y)\colon x,y\in F^{n}\}.}$

Let M₀, M₁, ..., M_{qⁿ - 1} be n × n matrices over F with the property that M_i – M_j is nonsingular whenever i ≠ j. For i = 0, 1, ...,qⁿ – 1 define,

${\displaystyle V_{i}=\{(x,xM_{i})\colon x\in F^{n}\},}$

usually referred to as the subspaces "y = xM_i". Also define:

${\displaystyle V_{q^{n}}=\{(0,y)\colon y\in F^{n}\},}$

the subspace "x = 0".

The set {V₀, V₁, ..., V_qⁿ} is a spread of V.

The matrices M_i used in this construction are called spread matrices or slope matrices.

Examples of regular spreads

A regular spread may be constructed in the following way. Let F be a field and E an n-dimensional extension field of F. Let V = E² considered as a 2n-dimensional vector space over F. The set of all 1-dimensional subspaces of V over E (and hence, n-dimensional over F) is a regular spread of V.

In the finite case, the field E = GF(qⁿ) can be represented as a subring of the n × n matrices over F = GF(q). With respect to a fixed basis of E over F, the multiplication maps, x → αx for α in E, are F-linear transformations and can be represented by n × n matrices over F. These matrices are the spread matrices of a regular spread.[28]

As a specific example, the following nine matrices represent GF(9) as 2 × 2 matrices over GF(3) and so provide a spread set of AG(2, 9).

${\displaystyle \left[{\begin{matrix}0&0\\0&0\end{matrix}}\right],\left[{\begin{matrix}1&0\\0&1\end{matrix}}\right],\left[{\begin{matrix}2&0\\0&2\end{matrix}}\right],\left[{\begin{matrix}0&1\\2&0\end{matrix}}\right],\left[{\begin{matrix}1&1\\2&1\end{matrix}}\right],\left[{\begin{matrix}2&1\\2&2\end{matrix}}\right],\left[{\begin{matrix}0&2\\1&0\end{matrix}}\right],\left[{\begin{matrix}1&2\\1&1\end{matrix}}\right],\left[{\begin{matrix}2&1\\2&2\end{matrix}}\right].}$

Modifying spread sets

The set of transversals of a regulus R also form a regulus, called the opposite regulus of R. If a spread S of PG(3, q) contains a regulus R, the removal of R and replacing it by its opposite regulus produces a new spread S^∗. This process is a special case of a more general process called derivation or net replacement.[29]

Starting with a regular spread of PG(3, q) and deriving with respect to any regulus produces a Hall plane. In more generality, the process can be applied independently to any collection of reguli in a regular spread, yielding a subregular spread;[30] the resulting translation plane is called a subregular plane. The André planes form a special subclass of subregular planes, of which the Hall planes are the simplest examples.

Notes

1. Eric Moorhouse has performed extensive computer searches to find projective planes. For order 25, Moorhouse has found 193 projective planes, 180 of which can be obtained from a translation plane by iterated derivation and/or dualization. For order 49, the known 1349 translation planes give rise to more than 309,000 planes obtainable from this procedure.
2. Geometry Translation Plane Retrieved on June 13, 2007
3. Hughes & Piper 1973, p. 100
4. Johnson, Jha & Biliotti 2007, p. 5
5. Hall 1943
6. There are many ways to coordinatize a translation plane which do not yield a quasifield, since the planar ternary ring depends on the quadrangle on which one chooses to base the coordinates. However, for translation planes there is always some coordinatization which yields a quasifield.
7. Dembowski 1968, p. 128. Note that quasifields are technically either left or right quasifields, depending on whether multiplication distributes from the left or from the right (semifields satisfy both distributive laws). The definition of a quasifield in Wikipedia is a left quasifield, while Dembowski uses right quasifields. Generally this distinction is elided, since using a chirally "wrong" quasifield simply produces the dual of the translation plane.
8. André 1954
9. Bruck & Bose 1964
10. Bruck & Bose 1964, p. 97
11. This notion generalizes that of a classical regulus, which is one of the two families of ruling lines on a hyperboloid of one sheet in 3-dimensional space
12. Bruck & Bose, p. 163 harvnb error: no target: CITEREFBruckBose (help)
13. Bruck & Bose, p. 164, Theorem 12.1 harvnb error: no target: CITEREFBruckBose (help)
14. Knuth 1965, p. 541
15. "Projective Planes of Small Order". ericmoorhouse.org. Retrieved 2020-11-08.
16. "Projective Planes of Order 16". ericmoorhouse.org. Retrieved 2020-11-08.
17. Reifart 1984
18. "Projective Planes of Order 25". ericmoorhouse.org. Retrieved 2020-11-08.
19. Dover 2019
20. Czerwinski & Oakden harvnb error: no target: CITEREFCzerwinskiOakden (help)
21. "Projective Planes of Order 27". ericmoorhouse.org. Retrieved 2020-11-08.
22. Dempwolff 1994
23. "Projective Planes of Order 32". ericmoorhouse.org. Retrieved 2020-11-08.
24. Mathon & Royle 1995
25. "Projective Planes of Order 49". ericmoorhouse.org. Retrieved 2020-11-08.
26. McKay, Royle & 2014. This is a complete count of the 2-dimensional non-Desarguesian translation planes; many higher-dimensional planes are known to exist. harvnb error: no target: CITEREFMcKayRoyle2014._This_is_a_complete_count_of_the_2-dimensional_non-Desarguesian_translation_planes;_many_higher-dimensional_planes_are_known_to_exist. (help)
27. Moorhouse 2007, p. 13
28. Moorhouse 2007, p. 15
29. Johnson, Jha & Biliotti 2007, p. 49
30. Bruck 1969

References

• André, Johannes (1954), "Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe", Mathematische Zeitschrift, 60: 156–186, doi:10.1007/BF01187370, ISSN 0025-5874, MR 0063056, S2CID 123661471
• Ball, Simeon; John Bamberg; Michel Lavrauw; Tim Penttila (2003-09-15), Symplectic Spreads (PDF), Polytechnic University of Catalonia, retrieved 2008-10-08
• Bruck, R.H. (1969), R.C.Bose and T.A. Dowling (ed.), "Construction Problems of finite projective planes", Combinatorial Mathematics and Its Applications, Univ. of North Carolina Press, pp. 426–514
• Bruck, R. H.; Bose, R. C. (1966), "Linear Representations of Projective Planes in Projective Spaces" (PDF), Journal of Algebra, 4: 117–172, doi:10.1016/0021-8693(66)90054-8
• Bruck, R. H.; Bose, R. C. (1964), "The Construction of Translation Planes from Projective Spaces" (PDF), Journal of Algebra, 1: 85–102, doi:10.1016/0021-8693(64)90010-9
• Czerwinski, Terry; Oakden, David (1992). "The translation planes of order twenty-five". Journal of Combinatorial Theory, Series A. 59 (2): 193–217. doi:10.1016/0097-3165(92)90065-3.
• Dembowski, Peter (1968), Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York: Springer-Verlag, ISBN 3-540-61786-8, MR 0233275
• Dempwolff, U. (1994). "Translation planes of order 27". Designs, Codes and Cryptography. 4 (2): 105–121. doi:10.1007/BF01578865. ISSN 0925-1022. S2CID 12524473.
• Dover, Jeremy M. (2019-02-27). "A genealogy of the translation planes of order 25". arXiv:1902.07838 [math.CO].
• Hall, Marshall (1943), "Projective planes" (PDF), Trans. Amer. Math. Soc., 54 (2): 229–277, doi:10.2307/1990331, JSTOR 1990331
• Hughes, Daniel R.; Piper, Fred C. (1973), Projective Planes, Springer-Verlag, ISBN 0-387-90044-6
• Johnson, Norman L.; Jha, Vikram; Biliotti, Mauro (2007), Handbook of Finite Translation Planes, Chapman&Hall/CRC, ISBN 978-1-58488-605-1
• Knuth, Donald E. (1965), "A Class of Projective Planes" (PDF), Transactions of the American Mathematical Society, 115: 541–549, doi:10.2307/1994285, JSTOR 1994285
• Lüneburg, Heinz (1980), Translation Planes, Berlin: Springer Verlag, ISBN 0-387-09614-0
• Mathon, Rudolf; Royle, Gordon F. (1995). "The translation planes of order 49". Designs, Codes and Cryptography. 5 (1): 57–72. doi:10.1007/BF01388504. ISSN 0925-1022. S2CID 1925628.
• McKay, Brendan D.; Royle, Gordon F. (2014). "There are 2834 spreads of lines in PG(3,8)". arXiv:1404.1643 [math.CO].
• Moorhouse, Eric (2007), Incidence Geometry (PDF), archived from the original (PDF) on 2013-10-29
• Reifart, Arthur (1984). "The classification of the translation planes of order 16, II". Geometriae Dedicata. 17 (1). doi:10.1007/BF00181513. ISSN 0046-5755. S2CID 121935740.
• Sherk, F. A.; Pabst, Günther (1977), "Indicator sets, reguli, and a new class of spreads" (PDF), Canadian Journal of Mathematics, 29 (1): 132–54, doi:10.4153/CJM-1977-013-6

Further reading

• Mauro Biliotti, Vikram Jha, Norman L. Johnson (2001) Foundations of Translation Planes, Marcel Dekker ISBN 0-8247-0609-9 .

External links

• Lecture Notes on Projective Geometry
• Publications of Keith Mellinger